BASICS:
  i = √(-1) | i² = -1 | i³ = -i | i⁴ = 1
  z = a + ib | |z| = √(a²+b²) | arg(z) = tan⁻¹(b/a)
  Conjugate: z̄ = a - ib | |z|² = z·z̄
  
ARGAND PLANE GEOMETRY:
  |z - z₁| = |z - z₂|: perpendicular bisector of segment z₁z₂
  |z - z₁| = r: circle with centre z₁ and radius r
  |z - z₁| + |z - z₂| = const > |z₁-z₂|: ellipse with foci z₁,z₂
  
DE MOIVRE'S THEOREM:
  (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)
  Use for: nth roots, trigonometric expansions
  
CUBE ROOTS OF UNITY:
  1, ω, ω² where ω = e^(2πi/3) = (-1+i√3)/2
  Properties: 1 + ω + ω² = 0 | ω³ = 1 | |ω| = 1

CONDITIONAL PROBABILITY:
  P(A|B) = P(A∩B)/P(B) where P(B) ≠ 0

BAYES' THEOREM:
  P(Aᵢ|B) = P(B|Aᵢ)P(Aᵢ) / Σ[P(B|Aⱼ)P(Aⱼ)]
  Use when: finding which event caused observed outcome

RANDOM VARIABLES:
  E(X) = Σ xᵢP(xᵢ) | Var(X) = E(X²) - [E(X)]²

BINOMIAL DISTRIBUTION:
  X ~ B(n,p): P(X=r) = ⁿCᵣ pʳ (1-p)^(n-r)
  E(X) = np | Var(X) = np(1-p)

IMPORTANT JEE PROBLEM TYPES:
  Cards/balls: without replacement = decreasing total
  Geometric probability: ratio of favourable region to total region
  Conditional: check if events are mutually exclusive vs independent
  Multinomial: distributing unlike/like objects into groups

VECTORS:
  Dot product: a·b = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃
  Cross product: |a×b| = |a||b|sinθ (magnitude = area of parallelogram)
  Triple product: a·(b×c) = volume of parallelepiped
  
3D GEOMETRY:
  Line through point (x₁,y₁,z₁) with direction (a,b,c):
    (x-x₁)/a = (y-y₁)/b = (z-z₁)/c
  
  Plane ax+by+cz+d = 0:
    Distance from (x₁,y₁,z₁): |ax₁+by₁+cz₁+d|/√(a²+b²+c²)
    Normal vector: (a,b,c)
  
  Angle between line and plane:
    sin θ = |l·n|/(|l||n|) where l = direction of line, n = normal to plane
  
  Shortest distance between skew lines:
    SD = |(b₁-a₁)·(d₁×d₂)| / |d₁×d₂|